Masses Mass Genesis Open Propagation Canonical Open Null Mode

The open null carrier

In physics, a photon has zero rest mass, which is why it always travels at the speed of light. The Recognition Science framework, which models reality as a ledger of discrete recognition events, needed a way to represent this zero rest mass within its own terms. The declaration canonicalOpenNullMode supplies that representation: it is a recognition mode (a pattern of events the ledger records) that propagates along an infinite ray of natural numbers, never looping back on itself.

The mode has two defining properties. First, its total recognition cost is zero, which the framework reads as zero rest mass. Second, the ray it travels along is injective, meaning each step goes to a new position, so the mode never repeats and therefore never closes into a finite loop. This matters because the framework's finite matter patterns, the ones with mass, close after eight ticks of a cycle; an open ray is what keeps light distinct from matter in the model.

The declaration also proves that such a mode exists. It constructs the canonical example: the zero-cost mode on the identity ray, where each natural number maps to itself. A companion theorem confirms that no open null mode can have a positive period, so it cannot be mistaken for a finite closed matter pattern. The framework's machine-checked library of formal theorems records all of this as a certificate: there is at least one zero-rest-mass carrier, the canonical one has zero rest mass, and none of them are periodic.

What the declaration does not claim is equally important. It establishes only the carrier for zero rest mass. Polarization and photon momentum are separate physics layers, named as such in the documentation, and the framework does not derive them here. The declaration also does not claim that this mode is unique, only that a canonical one exists. It is a building block, not a complete theory of light.

/-- The canonical open null carrier: canonical zero-cost NRM on the identity
ray. -/
def canonicalOpenNullMode : OpenNullLightMode where
  mode := Physics.NullRecognitionMode.canonicalNRM
  zero_cost := Physics.NullRecognitionMode.nrm_totalCost_zero
  ray := fun n => n
  ray_injective := by
    intro a b h
    exact h

/-- An injective ray cannot close with any positive period. -/
theorem openNullRay_no_positive_period
    (γ : OpenNullLightMode) (n p : ℕ) (hp : 0 < p) :
    γ.ray (n + p) ≠ γ.ray n := by
  intro h
  have hidx : n + p = n := γ.ray_injective h
  omega

theorem openNullLightMode_nonempty :
    ∃ γ : OpenNullLightMode, openNullRestMass γ = 0 :=
  ⟨canonicalOpenNullMode, openNullRestMass_zero canonicalOpenNullMode⟩

What this page does not claim

The declaration does not derive polarization or photon momentum. The declaration does not claim the open null mode is unique. The declaration does not establish that light is the only zero-rest-mass carrier.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/MassGenesis/OpenPropagation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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